Cosine transforf

ABSTRACT

COSE TRANSFORMS DEFINED, THE FOURIER TRANSFORM AT ANY FREQUENCY MAY BE COMPUTED.   THE FOURIER TRANSFORM OF A NON-PERIODIC WAVE IS CALCULATED BY PROGRAMMED COMPUTER USING MATRIX MULTIPLICATION. THE NON-PERIODIC WAVE TO BE TRANSFORMED IS CONVERTED TO A STAIRCASE WAVEFORM WHERE THE AMPLITUDE OF EACH STEP IN THE STAIRCASE WAVEFORM IS THE AVERAGE AMPLITUDE OF THE ORIGINAL WAVE. WALSH COEFFICIENTS OF THE STAIRCASE WAVEFORM ARE THEN COMPUTED BY MATRIX MULTIPLICATION OF THE WALSH MATRIX TIMES SAMPLE VALUES FOR THE WAVE WHERE THE SAMPLE VALUES ARE THE VARIOUS AMPLITUDES OF THE STAIRCASE WAVE DURING SEPARATE STEP OR SAMPLE INTERVALS. THEREAFTER, THE CONTINUOUS FOURIER TRANSFORM OF A NON-PERIODIC WAVE IS CALCULATED BY COMPUTING THE FOURIER TRANSFORM OF THE WALSH FUNCTIONS. THE LATTER COMPUTATION IS SIMPLIFIED BY MULTIPLYING THE WALSH COEFFICIENTS BY A SINCE ARRAY AND A COSINC (DEFINED HEREINAFTER) ARRAY. A FURTHER AND ULTIMATE SIMPLIFICATION OF THE ENTIER PROCESS IS ACCOMPLISHED BY MULTIPLYING THE WALSH MATRIX BY AN INTERMINGLING OF THE SINC AND COSINC ARRAYS. THE RESULT IS A MATRIX CONSISTING MOSTLY OF ZEROES AND HEREINAFTER DEFINED AS THE SINC MATRIX. MULTIPLICATION OF THE INPUT WAVE VALUES BY THE SINC MATRIX WILL YIELD DIRECTLY SINE/COSINE COEFFICIENTS OF THE FOURIER SINE/COSINE TRANSFORMS FOR THE STAIRCASE WAVEFORM THAT SIMULATES THE INPUT WAVE WITH THE COEFFICIENTS OF THE FOURIER SINE/

DEFENSIVE PUBLICATION UNITED STATES PATENT OFFICE Published at the request of the applicant or owner in accordance with the Notice of Dec. 16, 1969, 869 O.G. 687. The abstracts of Defensive Publication applications are identified by distinctly numbered series and are arranged chronologically. g The heading of each abstract indicates the number of pages of specification. including claims and sheets of drawings contained in the application as originally filed. The files of these applications are available to the public for inspectionand reproduction may be purchased for 30 cents a sheet.

Defensive Publication applications have not been examined as to the merits of alleged invention. The Patent Oflice makes 1 no assertion as to the novelty of the disclosed subject matter.

PUBLISHED MAY 8, 1973 T910,009 The Fourier transform of a non-periodic wave is cal- SINC/COSINC TRANSFORM-A CONTINUOUS culated by programmed computer using matrix multipli- FOURIER TRANSFORM cation. The non-periodic wave to be transformed is conl APramson, f asslgnor to Interim verted to a staircase waveform where the amplitude of t? g gg ag g g g gg each step in the staircase waveform is the average amplid? 3 4 tude of the original wave. Walsh coefiicients of the stair- CL case Waveform are then computed by matrix multiplica- 10 Sheets Drawing. 57 Pages Specification tion of the Walsh matrix times sample values for the I i wave where the sample values are the various amplitudes CDMBINE COEF of the staircase Wave during separate step or sample inn lglun 72 WITH smE tervals. Thereafter, the continuous Fourier transform of mm i A AND 1 a non-periodic wave is calculated by computing the Fou- P P Q Q' g COMPUTE rier transform of the Walsh functions. The latter coml gg t P u I putation is simplified by multiplying the Walsh coefli- 52 L 1 1 cients by a sine array and a cosine (defined heremafter) POINTSVECTOR array. A further and ultimate simplification of the en- E tire process is accomplished by multiplying the Walsh matrix by an intermingling of the sine and cosine arrays. L The result is a matrix consisting mostly of zeroes and l hereinafter defined as the sine matrix. Multiplication of ggactm 5 the input wave values by the sine matrix will yield di- SET TRF T00 rectly sine/ cosine coeflicients of the Fourier sine/cosine l l I l ZERO FREMERM i transforms for the staircase Waveform that simulates the CALCULATEAND input wave. With the coefiicients of the Fourier sine/ L ADDZERO 85 FREQTERM cosine transforms defined, the Fourier transform at any frequency may be computed.

May R, Rm

SINC/GOSINC TRANSFORM A CONTINUOUS FOURIE-R TRANSFORM Filed July 5.

R. F ABRAMSON 'RRRRRRR 10 Sheets-Sheet 1.

PERIOD A ,A A AVC. AMPLITUDE DURING EACH SAMPLING INTERVAL= EREO'S.

OF FOURIER TRANSFORM IS CALCULATED 1972 HQ. MA

GET

T BASE PERIOD INTERVAL OF INTEREST) N ---NUMBEROFSAMPLINCINTERVALS IN BASE AT RRlcR REAL AND IMAGINARY PART COMPUTE SIN RCOS COEF. FOR FOURIER TRANSFORM OF COMPUTED WALSH FUNCTIONS COMPUTE REAL R IMAGINARY PART OF FOURIER TRANSFORM COMPUTE SIM R COS COEF.

FOR FOURIER TRANSFORM OF T Y MATRIX OE SIMC MATRIX a, ma

R- F'- ABRAMSON SINC/COSINC TRANSFORM A CONTINUOUS-FOURIER TRANSFORM Filed July :5, 1972 FROAI 55 TII IE v MATRIX IIULTIPLICATIOIII INPUT VECTOR XX SIRC I IATRIX FOURIER COEFFICIENTS TO TI l0 Sheets-Sheet E FREQUENCY l I I (III FREQUENCY AIf) AIf)

FROM 55 FROM 95 MATRIX MULTIPLICATION INPUT VECTOR WALSH MATRIX? WALSH COEFFICIENTS MATRIX I IULTIPLICATION WALSH COEFFICIENTS I IIITERI IINCLEO SINCI' COSIRC ARRAY FOURIER COEFFICIENTS TO TI May 8, 1973 R. F. ABRAMSON I SINC/COSINC TRANSFORM A CONTINUOUS FOURIER TRANSFORM Filed Jul :5, 1.972

10 Sheets-Sheet 8 35:35 2: l L a E: :l T wvl/g u, x P #5 i T NM 5 T l g g a w 03 8, M0 R. F. ABRAMSON T10,

S'INC/COSINC TRANSFORM A CONTINUOUS F OURIER TRANSFORM Filed July 5, 1072 10 Sheets-Sheet A I I I WAL (0,0)

I CAL (1,0) I I -I L SAL (2,0

I W CAL (2,0 I

W SAL (5,0)

. 1 CAL (5,0)

SAL (4,0)

CAL (5,9)

I I I I SAL (0,0)

SAL (1,8)

' SAL (5,0) I

May 8 1%73 ABRAMSQN 1919 099 SINC/COSINC TRANSFORM A CONTINUOUS FOURIER TRANSFORM Fil iy 1972 1o Sheets-Sheet 8 PEG, L5G

L L GMPRD GMPRD /452 L COEF WAVE WAVE .454 smcmmmx WALSH MATRIX= WALSH GOEF GMPRD F860 L5 smc/cosmc ARRAY= L FOURLERCOEF L GMPRD 158 L WAVHWJALSH =SAL&CAL COEFFICIENTS COLLECT CAL COLLECT SAL COEFFICIENTS COEFFICIENTS GMPRD CAL COEFS SING ARRAY =SINE COEF'S SAL COEF'S COSLNC ARRAY L-OS COEFS 0 W73 R. F. ABRAMSON SINC/COSINC TRANSFORM A CONTINUOUS FOUFIIER TRANSFORM Filed July 5, 1972 l0 Sheets-Sheet 9 coIuNN INDEX I 8 0FS|NCMATRlX 0425456789 IND-EX coLuNIN INDEX- 0F INPUT VECTOR [o 'I 557155 oo] INPUTVECTOR may 8 1973 F. ABRAMSON T910909 SINC/COSINC TRANSFORM A CONTINUOU5.FOURIER TRANSFORM l0 Sheets-Sheet 10 Filed July 5, 1972 PRINT FREQ (M), REAL, IMAG REAL=IREAL COEF (1-1) *1/2 

